<form action="" method="post" name="booking_form" class="booking" style="width: auto">
<select name="customer_name" onChange="???????">
<option value="index.php">Click to select your school </option>
<?php

// Make a MySQL Connection

$username="";

$password="";

$database="";

mysql_connect('localhost',$username,$password);

@mysql_select_db($database) or die( "Unable to select database");

$query = "SELECT * FROM table";
$result = mysql_query($query) or die(mysql_error());
while($row = mysql_fetch_array($result)){

$id = $row['id'];
$school = $row['customer_name'];
$phone = $row['customer_phone'];
$address = $row['customer_address'];
$city = $row['customer_city'];
$state = $row['customer_state'];
$zip = $row['customer_zip'];
$notes = $row['customer_notes'];
?>
<option value="<?php echo $school; ?>"><?php echo "$school".", "."$city"; ?></option>
<?php }

?></select>
<input type="text" name="customer_address">
<input type="text" name="customer_city">
<input type="text" name="customer_state">
<input type="text" name="customer_zip">
</form>

<select name="customer_name">
<option id='selects' value="index.php">Click to select your school </option>

$("select[name='customer_name']").change(function(){
            $.post("-YOUR PHP FILE-", customer_name: $(this).val(), function(data){
               schools = data.split(",");
               for(var i in schools){
               $("select[name='customer_name']").append("<option>"+schools[i]+"</option>");
               }
            })
       })

<?php
          //You connected and got an array contains your school names($schools)
          $schools = implode(",",$schools);
          echo $schools;
       ?>

var schoolInfo = { id1 : {name:"Name 1", address:"address line" ...},
id2: {} .... }

<option value="school" sid="id1">school, city</option>

.change(function(){
     var sid = $(this).attr('sid');
     $("input[name='customer_address']").val(schoolInfo[sid].address);
     $("input[name='customer_city']").val(schoolInfo[sid].city);
     $("input[name='customer_zip']").val(schoolInfo[sid].zip);
});